Difference between revisions of "2011 AMC 12A Problems/Problem 18"

(Solution 2)
(Solution 1)
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Plugging in some values, we see that the  graph of the equation <math>|x+y|+|x-y| = 2</math> is a square bounded by <math>x= \pm 1</math> and <math>y = \pm 1</math>.
 
Plugging in some values, we see that the  graph of the equation <math>|x+y|+|x-y| = 2</math> is a square bounded by <math>x= \pm 1</math> and <math>y = \pm 1</math>.
  
Notice that <math>x^2 - 6x + y^2 = (x-3)^2 + y^2 - 9</math> means the square of the distance from a point <math>(x,y)</math> to point <math>(3,0)</math> minus 9. To maximize that value, we need to choose the point in the feasible region farthest from point <math>(3,0)</math>, which is <math>(-1, \pm 1)</math>. Either one, when substituting into the function, yields <math>8 \rightarrow \boxed{(D)}</math>.
+
Notice that <math>x^2 - 6x + y^2 = (x-3)^2 + y^2 - 9</math> means the square of the distance from a point <math>(x,y)</math> to point <math>(3,0)</math> minus 9. To maximize that value, we need to choose the point in the feasible region farthest from point <math>(3,0)</math>, which is <math>(-1, \pm 1)</math>. Either one, when substituting into the function, yields <math>\boxed{\textbf{(D)}\ 8}</math>.
  
 
== Solution 2 ==
 
== Solution 2 ==

Revision as of 21:49, 2 November 2021

Problem

Suppose that $\left|x+y\right|+\left|x-y\right|=2$. What is the maximum possible value of $x^2-6x+y^2$?

$\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

Solution 1

Plugging in some values, we see that the graph of the equation $|x+y|+|x-y| = 2$ is a square bounded by $x= \pm 1$ and $y = \pm 1$.

Notice that $x^2 - 6x + y^2 = (x-3)^2 + y^2 - 9$ means the square of the distance from a point $(x,y)$ to point $(3,0)$ minus 9. To maximize that value, we need to choose the point in the feasible region farthest from point $(3,0)$, which is $(-1, \pm 1)$. Either one, when substituting into the function, yields $\boxed{\textbf{(D)}\ 8}$.

Solution 2

Since the equation $|x+y|+|x-y| = 2$ is dealing with absolute values, the following could be deduced: $(x+y)+(x-y)=2$,$(x+y)-(x-y)=2$, $-(x+y)+(x-y)=2$, and $-(x+y)-(x-y)=2$. Simplifying would give $x=1$, $y=1$, $y=-1$, and $x=-1$. In $x^2-6x+y^2$, we care most about $-6x,$ since both $x^2$ and $y^2$ are non-negative. To maximize $-6x$, though, $x$ would have to be -1. Therefore, when $x=-1$ and $y=-1$ or $y=1$, the equation evaluates to $\boxed{(D)}$ $8$.

See also

2011 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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